Explanation Given: The given function is f ( x , y ) = 5 x y − 7 x 2 − y 2 + 3 x − 6 y − 4 . Approach: “For a function z = f ( x , y ) , let f x x , f y y , and f x y all exist in a circular region contained in the xy -plane with center ( a , b ) . Further, let f x ( a . b ) = 0 and f y ( a . b ) = 0 . Define the number D , known as the discriminant, by D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 Then, a. f ( a , b ) is a relative maximum if D > 0 and f x x ( a , b ) < 0 . b. f ( a , b ) is a relative minimum if D > 0 and f x x ( a , b ) > 0 . c. f ( a , b ) is a saddle point (neither a maximum nor a minimum) if D < 0 . d. if D = 0 , the test gives no information.” Calculation: Consider the given function, f ( x , y ) = 5 x y − 7 x 2 − y 2 + 3 x − 6 y − 4 . Find the values f x ( x , y ) , and f y ( x , y ) : f x ( x , y ) = ∂ ∂ x ( 5 x y − 7 x 2 − y 2 + 3 x − 6 y − 4 ) = 5 y − 14 x + 3 and, f y ( x , y ) = ∂ ∂ y ( 5 x y − 7 x 2 − y 2 + 3 x − 6 y − 4 ) = 5 x − 2 y − 6 A function has relative extrema at a point, if f x ( x , y ) = 0 5 y − 14 x + 3 = 0 And if f y ( x , y ) = 0 ,

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 9.3, Problem 8E

To determine

To find:

All the points for relative extrema and identify any saddle point for the given function.

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Chapter 9.3, Problem 7E

Chapter 9.3, Problem 9E

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Chapter 9 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

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All the points for relative extrema and identify any saddle point for the given function.

Solution Summary: The author explains that the function has a relative maximum at the point (-8,-23).

To find: All the points for relative extrema and identify any saddle point for the given function.

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Chapter 9 Solutions

To find:

All the points for relative extrema and identify any saddle point for the given function.